John Horton Conway (1937–2020) in 2009 [Photo (c) Denise Applewhite, Princeton University]
Continue reading ‘Amusical Permutations and Unsettleable Problems’
Posts Tagged 'Logic'
Amusical Permutations and Unsettleable Problems
Published April 6, 2023 Occasional ClosedTags: Arithmetic, Logic, Number Theory
Sets that are Elements of Themselves: Verboten
Published March 9, 2023 Occasional ClosedTags: Logic, Set Theory
Russell’s Paradox.
Can a set be an element of itself? A simple example will provide an answer to this question. Continue reading ‘Sets that are Elements of Themselves: Verboten’
Goldbach’s Conjecture and Goldbach’s Variation
Published July 14, 2022 Occasional ClosedTags: Logic, Number Theory
Goldbach’s Conjecture is one of the great unresolved problems of number theory. It simply states that every even natural number greater than two is the sum of two prime numbers. It is easily confirmed for even numbers of small magnitude.
The conjecture first appeared in a letter dated 1742 from German mathematician Christian Goldbach to Leonhard Euler. The truth of the conjecture for all even numbers up to four million million million (
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Hyperreals and Nonstandard Analysis
Published March 10, 2022 Occasional ClosedTags: Analysis, Logic
Following the invention of calculus, serious concerns persisted about the mathematical integrity of the method of infinitesimals. Leibniz made liberal use of infinitesimals, with great effect, but his reasoning was felt to lack rigour. The Irish bishop George Berkeley criticised the assumptions underlying calculus, and his objections were not properly addressed for several centuries. In the 1800s, Bolzano, Cauchy and Weierstrass developed the 
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Topsy-turvy Maths: Proving Axioms from Theorems
Published November 4, 2021 Irish Times ClosedTags: Logic, Philosophy
Mathematics is distinguished from the sciences by the freedom it enjoys in choosing basic assumptions from which consequences can be deduced by applying the laws of logic. We call the basic assumptions axioms and the consequent results theorems. But can things be done the other way around, using theorems to prove axioms? This is a central question of reverse mathematics [TM222 or search for “thatsmaths” at irishtimes.com].
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Goldbach’s Conjecture: if it’s Unprovable, it must be True
Published March 4, 2021 Irish Times ClosedTags: Euler, Logic, Number Theory
Image © easycalculation.com
The starting point for rigorous reasoning in maths is a system of axioms. An axiom is a statement that is assumed, without demonstration, to be true. The Greek mathematician Thales is credited with introducing the axiomatic method, in which each statement is deduced either from axioms or from previously proven statements, using the laws of logic. The axiomatic method has dominated mathematics ever since [TM206 or search for “thatsmaths” at irishtimes.com].
Continue reading ‘Goldbach’s Conjecture: if it’s Unprovable, it must be True’
The phrase `squaring the circle’ generally denotes an impossible task. The original problem was one of three unsolved challenges in Greek geometry, along with trisecting an angle and duplicating a cube. The problem was to construct a square with area equal to that of a given circle, using only straightedge and compass.
Georg Cantor developed a remarkable theory of infinite sets. He was the first person to show that not all infinite sets are created equal. The number of elements in a set is indicated by its cardinality. Two sets with the same cardinal number are “the same size”. For two finite sets, if there is a one-to-one correspondence — or bijection — between them, they have the same number of elements. Cantor extended this equivalence to infinite sets.
Mathematics and the Nature of Physical Reality
Published October 1, 2020 Irish Times ClosedTags: Logic, Philosophy
Applied mathematics is the use of maths to address questions and solve problems outside maths itself. Counting money, designing rockets and vaccines, analysing internet traffic and predicting the weather all involve maths. But why does this work? Why is maths so successful in describing physical reality? How is it that the world can be understood mathematically? [TM196, or search for “thatsmaths” at irishtimes.com].
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Berry’s Paradox and Gödel’s Incompleteness Theorem
Published July 30, 2020 Occasional ClosedTags: Logic
Left: Argentine-American mathematician
Gregory Chaitin [image from here]. Right: American philosopher and logician
George Boolos [image Wikimedia Commons].
THE SMALLEST NATURAL NUMBER THAT CANNOT BE
DEFINED IN FEWER THAN TWENTY WORDS.
Continue reading ‘Berry’s Paradox and Gödel’s Incompleteness Theorem’
Jean Buridan (c. 1300-1360).
“Buridan’s Ass” is a paradox in philosophy, in which a hungry donkey, located at the mid-point between two bales of hay, is frozen in indecision about which way to go and faces starvation — he is unable to move one way or the other.
Jean Buridan was a French philosopher who lived in the fourteenth century. He was not interested in donkeys, but in human morality. He wrote that if two courses of action are judged to be morally equal, we must suspend a decision until the right course of action becomes clear. The idea of the paradox can be found in the writings of the ancients, including Aristotle.
Ross-Littlewood Paradox [Image from Steemit website: here. ]
The “Napoleon of Crime” and The Laws of Thought
Published November 15, 2018 Irish Times ClosedTags: History, Logic
A fascinating parallel between a brilliant mathematician and an arch-villain of crime fiction is drawn in a forthcoming book – New Light on George Boole – by Des MacHale and Yvonne Cohen. Professor James Moriarty, master criminal and nemesis of Sherlock Holmes, was described by the detective as “the Napoleon of crime”. The book presents convincing evidence that Moriarty was inspired by Professor George Boole [TM151, or search for “thatsmaths” at irishtimes.com].
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“Dividends and Divisors Ever Diminishing”
Published June 14, 2018 Occasional ClosedTags: Arithmetic, Logic
Next Saturday is Bloomsday, the anniversary of the date on which the action of Ulysses took place. Mathematical themes occur occasionally throughout Ulysses, most notably in the penultimate episode, Ithaca, where the exchanges between Leopold Bloom and Stephen Dedalus frequently touch on weighty scientific matters. [Last week’s ThatsMaths post]
Joyce in Zurich: did he meet Zermelo?
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Children sometimes amuse themselves searching for the biggest number. After trying millions, billions and trillions, they realize that there is no end to the game: however big a number may be, we can always add 1 to produce a bigger number: the set of counting numbers is infinite. The concept of infinity has intrigued philosophers since antiquity, and it leads to many surprises and paradoxical results [TM110 or search for “thatsmaths” at irishtimes.com].
The Shaky Foundations of Mathematics
Published December 1, 2016 Irish Times ClosedTags: Algorithms, Logic, Number Theory
The claim is often made that mathematical results are immutable. Once proven, they remain forever valid. But things are not so simple. There are problems at the very core of mathematics that cast a shadow of uncertainty. We can never be absolutely sure that the foundations of our subject are rock-solid [TM104 or search for “thatsmaths” at irishtimes.com].
Left: Plato and Aristotle. Centre: Pythagoras. Right: Euclid [Raphael, The School of Athens]
The ancient Greeks put geometry on a firm footing. Euclid set down a list of axioms, or basic intuitive assumptions. Upon these, the entire edifice of Euclidean geometry is constructed. This axiomatic approach has been the model for mathematics ever since.
The links between mathematics and music are manifold. Mathematics can be set to music in a simple but surprising manner. For the award ceremony of the Gödel Medal in 2014, a musical interpretation of Gödel’s incompleteness Theorems was written by Danish composer Niels Marthinsen. It encodes the basic axioms of number theory that form the focus of Gödel’s Theorems.
The Peano Axioms in symbolic form.
The Year of George Boole
Published December 4, 2014 Irish Times ClosedTags: Algorithms, Computer Science, History, Logic
This week’s That’s Maths column in The Irish Times (TM058, or search for “thatsmaths” at irishtimes.com) is about George Boole, the first Professor of Mathematics at Queen’s College Cork.
Many of us recall the sense of wonder we felt upon learning that there is no biggest number; for some of us, that wonder has never quite gone away. It is obvious that, given any counting number, one can be added to it to give a larger number. But the implication that there is no limit to this process is perplexing.
Georg Cantor (1845 – 1918) around 1870 (left) and in later life (right).
Invention or Discovery?
Published July 24, 2014 Occasional ClosedTags: Analysis, Logic, Number Theory, Social attitudes
Is mathematics invented or discovered? As many great mathematicians have considered this question without fully resolving it, there is little likelihood that I can provide a complete answer here. But let me pose a possible answer in the form of a conjecture:
Conjecture: Definitions are invented. Theorems are discovered.
The goal is to prove this conjecture, or to refute it. Below, some arguments in support of the conjecture are presented. Continue reading ‘Invention or Discovery?’
One of the most amazing and counter-intuitive results in mathematics was proved in 1924 by two Polish mathematicians, Stefan Banach and Alfred Tarski. Banach was a mathematical prodigy, and was the founder of modern functional analysis. Tarski was a logician, educated at the University of Warsaw who, according to his biographer, “changed the face of logic in the twentieth century” through his work on model theory.
ThatsMaths 